I have a problem with "If supply of workers is short of demand, price should go up". In a very basic view this makes sense. But not in even slightly more sophisticated models.

A company makes a certain "marginal" revenue off the employee. It's not always easy or even possible to compute the exact amount, but the salary is limited by this marginal revenue. The difference between marginal revenue and salary is the marginal profit. If that profit is too small, management might not be able to get the money to even hire this employee.

Supply and demand in the labor market is a very fickle thing, and I am certainly no expert. But adding job opportunities and subtracting immigrants is not going to fly.